Seismic methods at the long wavelength limit detect the total seismic velocity anomaly Xoff ?c where ?c is the change of the velocity of the recorded wave. If seismic velocity varies linearly with temperature over this range, the total velocity anomaly is proportional to ?TdownXdown. As a caveat, trace amounts of partial melt may well affect seismic velocity in the asthenosphere, so the coefficient to convert temperature variation to seismic velocity variation is not obvious. Linearization may even be inappropriate and the proportionality coefficient may vary with depth as the phase assemblage changes. Till et al. review these issuesplicated relationships between temperature and seismic velocity are beyond the scope of this paper.
van Wijk et al. presented models of downwellings at the edge of rifted thin lithosphere. Their models had T? of ?57 and ?107 K within the range where stagnant-lid formalism is applicable. Their viscosity increased with depth along an adiabat, increasing by a factor of e per scale depth D? of 83 km. Their downwellings widened with depth and then ponded ?300 km beneath the top of the asthenosphere. Their viscosity increased by a factor of 37 over that interval to approximately 10 20 Pa s, consistent with the results of Lee et al. and Harig et al. . From (15) the model heat circulate in equilibrium with thin lithosphere asiame coupons in the model of van Wijk et al. is about 2.5 times that in my models, so the anomaly from (15) is 2.5 times mine, 1750 K km.
van Hunen and Zhong discussed the effective length scale for flow in isoviscous asthenosphere. In agreement with my approach, they concluded that the depth beneath the base of the lithosphere provides a length scale and that the actual length scale varies modestly due to the vagaries of the spacing of downwellings. For completeness, the increase of viscosity with depth provides another length scale for the parameter Xflow. From (13b), velocity decreases by a factor of e over 2D? = 166 km in the calculations of van Wijk et al. , providing a length scale similar to that for depth below the lithosphere. Well down into the asthenosphere, the effective value of parameter Xflow is uncertain (in the absence of numerical calculations) by a factor of ?2, which is much less than the uncertainty in actual viscosity in the ratio Xdown/?down in (13b) and (14).
step 3.4. Temperature Examine Contained in this good Downwelling
Calibrated detail by detail tomography comes with the possible off resolving the warmth examine inside a good downwelling. We introduce easy scaling objections referred to numerical data.
The term (?down/Xflow) 1/2 /?Tdown 2 contains the most uncertain factors. First, the initial value of ?Tdown scales with T?. Geotherms scaled to given fractions of T? below the adiabat thus penetrate to greater depths at small T? = 40 K compared with large T? = 100 K (Figure 1). Second, the viscosity increased strongly with depth in the models of van Wijk et al. . Their downwellings to widened making conduction unimportant and allowing large temperature differences to persist well into downwellings.
Regarding seismology, the scaling dating signify initially downwellings (in which T
? is small) have small temperature contrasts and small total anomalies at a given laterally averaged heat flow. Depth-dependent viscosity is quite important, causing downwellings to widen with depth and become independent of the properties of the boundary layer in (14). A further effect, not modeled here, is that the thermal gradient at mid-mantle depths is subadiabatic [ Bunge, 2005 ]. The deep distal end of downwellings thus loses its identity from the ambient mantle and becomes undetectable [ Sleep, 2011 ].